the classification of clifford gates over qubits

The Classification of Clifford Gates over Qubits

Daniel Grier, Luke SchaefferMIT

Clifford Classification

Introduction

Section 1

Introduction

Classification

Study universality by finding non-universal gate sets.

Each non-universal set has some property (an invariant) which is an obstacle touniversality.

Sometimes non-universal sets are interesting in their own right. E.g., Clifford gates.

Definition

Let 〈G 〉 denote the set of unitaries constructible as circuits with gates from G .

Goal: Characterize all possible sets 〈G 〉.

Introduction

Classification

Definition

Introduction

Classification

Definition

Introduction

Classification

Definition

Introduction

Warm up: Reversible Gates

Definition

A gate is reversible if it maps each classical input to some classical output.

E.g., CNOT, the Toffoli gate, CSWAP.

Completely classified (Aaronson, Grier, S. 2017) in a classical circuit model.

We will consider a simplified version under a quantum circuit model.

Introduction

Warm up: Reversible Gates

Definition

A gate is reversible if it maps each classical input to some classical output.

E.g., CNOT, the Toffoli gate, CSWAP.

Completely classified (Aaronson, Grier, S. 2017) in a classical circuit model.

We will consider a simplified version under a quantum circuit model.

Introduction

Reversible Gate Classification

Theorem

Given a collection G of reversible gates, 〈G 〉 is one of the following six sets:

〈Toffoli〉,〈Fredkin〉,〈CNOT〉,〈T4〉, (where T4 is yet to be defined)

〈NOT〉, or

〈SWAP〉.

Introduction

Toffoli

T4 Fredkin

Introduction

Invariants

Observe that CNOT satisfies the following property.

Affine Invariant

Every output bit is an affine function (modulo 2) of the input bits.

Also, if all gates in a circuit satisfy this property, so does the entire circuit. (This is why it isan invariant)

Similarly, Fredkin satisfies the following property.

Conservativity Invariant

The Hamming weight of the input bits is the same the Hamming weight of the output bits.

Once again, if all gates have this property then so does the circuit.

Introduction

Invariants

Affine Invariant

Introduction

Invariants

Affine Invariant

Introduction

Obstacles to Universality

If all gates are affine, then the gate set is not universal.

If all gates are conservative, then the gate set is not universal.

On the other hand, the classification shows us that these are the only obstacles.

If at least one gate is not affine, and at least one gate is not conservative, then the gateset is universal.

Introduction

Obstacles to Universality

If all gates are affine, then the gate set is not universal.

If all gates are conservative, then the gate set is not universal.

On the other hand, the classification shows us that these are the only obstacles.

If at least one gate is not affine, and at least one gate is not conservative, then the gateset is universal.

Introduction

Clifford Group

Recall the Pauli matrices

(1 00 1

), X =

(0 11 0

(0 −ii 0

), Z =

(1 00 −1

The Pauli group on n qubits is

Pn = {±1,±i} × {I ,X ,Y ,Z}⊗n.

The Clifford group on n qubits is the set of unitaries that normalize the Pauli group.

Cn = {U : UPnU† = Pn}

Introduction

For example, CNOT ∈ C2.

CNOT ·(X ⊗ I ) · CNOT† = X ⊗ X

CNOT ·(I ⊗ X ) · CNOT† = I ⊗ X

CNOT ·(Z ⊗ I ) · CNOT† = Z ⊗ I

CNOT ·(I ⊗ Z ) · CNOT† = Z ⊗ Z

Similarly, H = 1√2

( 1 11 −1 ),S = ( 1 0

0 i ) ∈ C1.

HXH† = Z HZH† = X

SXS† = Y SZS† = Z

Fact: 〈CNOT,H, S〉 =⋃

Introduction

( 1 11 −1 ),S = ( 1 0

0 i ) ∈ C1.

Introduction

( 1 11 −1 ),S = ( 1 0

0 i ) ∈ C1.

Introduction

Classification of Clifford Gates

Theorem

Given a collection G of Clifford gates, 〈G 〉 is one of 57 possible classes.

57 sets is too many to list or draw on one slide.

30 classes are trivial, generated by single qubit gates.

27 more interesting classes

Introduction

Classification of Clifford Gates

Theorem

57 sets is too many to list or draw on one slide.

30 classes are trivial, generated by single qubit gates.

27 more interesting classes

Introduction

RX RY RZ

P + RX P + RY P + RZ

Γ−−+Γ−+−Γ+−−Γ+++ θY+Z θY−Z θX+Z θX−Z θX+Y θX−Y

P + Γ

X + θYZ Y + θXZ Z + θXY

θ+++ θ+−− θ−+− θ−−+

Introduction

C(Y , X ) + P + RX C(Z , Y ) + P + RY C(X , Z) + P + RZ

C(X , Z) + P T4 + P + RX C(Y , X ) + P T4 + P + RY C(Z , Y ) + P T4 + P + RZ

C(X , X ) + P + RX C(Y , Y ) + P + RY C(Z , Z) + P + RZ

C(X , X ) + P C(X , X ) + RX C(X , X ) + X + θYZ C(Y , Y ) + P C(Y , Y ) + RY C(Y , Y ) + Y + θXZ C(Z , Z) + P C(Z , Z) + RZ C(Z , Z) + Z + θXY

C(X , X ) + X C(Y , Y ) + Y C(Z , Z) + Z

T4 + P + Γ

T4 + P

Introduction

Invariants and Tableaux

We state invariants for Clifford gates based on the tableau of the gate.

Tableau Intuition

Clifford gates ' Linear transformationsTableaux ' Matrices

Matrix Representation

Fix a basis for the vector space.

Write the image of each basis element as a (linear) combination of basis elements.

Format data as a table.

Result: 2n × 2n boolean matrix, length 2n bit vector (for phase).

Introduction

Tableau Intuition

Matrix Representation

Fix a basis for the vector space.

Introduction

Tableau Intuition

Matrix Tableau Representation

Fix a basis for the vector space Pauli group.

Introduction

Tableau Intuition

Matrix Tableau Representation

Fix a basis for the vector space Pauli group.

Introduction

Tableau Examples

E.g., CSIGN and H have the following tableaux, ignoring phase bits.

CSIGN =

1 0 0 10 1 0 0

0 1 1 00 0 0 1

(0 11 0

CSIGN(X ⊗ I ) CSIGN† = (X ⊗ I )(I ⊗ Z ).

Basis is X ⊗ I ,Z ⊗ I , I ⊗ X , I ⊗ Z .

First row is 1, 0, 0, 1

Introduction

Tableau Examples

E.g., CSIGN and H have the following tableaux, ignoring phase bits.

CSIGN =

1 0 0 10 1 0 0

0 1 1 00 0 0 1

(0 11 0

CSIGN(X ⊗ I ) CSIGN† = (X ⊗ I )(I ⊗ Z ).

Basis is X ⊗ I ,Z ⊗ I , I ⊗ X , I ⊗ Z .

First row is 1, 0, 0, 1

Introduction

Invariant Example

CSIGN =

1 0 0 10 1 0 0

0 1 1 00 0 0 1

(0 11 0

Definition

A gate is Z -preserving if there is a 0 in the bottom left of every block of the tableau.

CSIGN is Z -preserving, H is not.

Introduction

Outline

Circuit Model

Quick Tour of the Lattice

Important gatesBroad structural description

Elements of the proof

Universal construction

Circuit Model

Section 2

Circuit Model

Axiom 1

Given circuits for U and V , we have circuits for UV and U ⊗ V .Composition

Tensor Product

Circuit Model

Axiom 2 (Permutation)

1 Does locality matter?• •

2 Does order matter?• • ×

• ×

Our Solution

Assume SWAP is in our gate set.

Circuit Model

Axiom 2 (Permutation)

1 Does locality matter?• •

2 Does order matter?• • ×

• ×

Our Solution

Assume SWAP is in our gate set.

Circuit Model

Axiom 3 (Ancillas)

Our ancilla policy allows

1 arbitrarily many ancilla qubits,

2 initialized to arbitrarily complex quantum states,

3 but they must be returned to original state.

Circuit Model

Axiom 3 (Ancillas)

Circuit Model

Axiom 3 (Ancillas)

Circuit Model

Ancilla Qubit Objections

In principle, couldn’t the ancilla states be impractical to construct?

Maybe, but they only need to be constructed once, then reused.

For this classification we use a discrete set of 1 and 2 qubit ancilla states.

Clifford gates and magic states are universal for quantum computation, aren’t they?

Yes, but only if the magic states are destroyed in the process, and our ancillas must bereturned intact.

Circuit Model

Tour of the Lattice

Section 3

Tour of the Lattice

Reminder

Theorem

The 57 classes break into two major groups:

30 classes generated by single qubit gates, and

27 other classes.

Tour of the Lattice

RX RY RZ

P + RX P + RY P + RZ

Γ−−+Γ−+−Γ+−−Γ+++ θY+Z θY−Z θX+Z θX−Z θX+Y θX−Y

P + Γ

X + θYZ Y + θXZ Z + θXY

θ+++ θ+−− θ−+− θ−−+

Tour of the Lattice

Understanding Single Qubit Classes

Each subgroup of C1 defines a class.

Each class generated by single qubit gates corresponds to a subgroup of C1.

C1 is isomorphic to the symmetries of the cube, or S4.

These have 30 subgroups, hence 30 classes.

Tour of the Lattice

Understanding Single Qubit Classes

Each subgroup of C1 defines a class.

Each class generated by single qubit gates corresponds to a subgroup of C1.

C1 is isomorphic to the symmetries of the cube, or S4.

These have 30 subgroups, hence 30 classes.

Tour of the Lattice

C(X , X ) + X C(Y , Y ) + Y C(Z , Z) + Z

T4 + P + Γ

T4 + P

Tour of the Lattice

Color Legend

The colors indicate the gate is X -, Y -, or Z -preserving.

Definition

A gate is Z -preserving if there is a 0 in the bottom left of every block of the tableau.

E.g., CSIGN, 1 0 0 10 1 0 0

0 1 1 00 0 0 1

Tour of the Lattice

Preservation invariants

There are similar invariants for X - and Y -preserving gates. The invariants impose thefollowing restrictions on a typical block ( a b

c d ).

X -preserving: b = 0,

Y -preserving: a + b + c + d = 0,

Z -preserving: c = 0.

So CSIGN is not X -preserving or Y -preserving.1 0 0 10 1 0 0

0 1 1 00 0 0 1

Tour of the Lattice

Preservation invariants

There are similar invariants for X - and Y -preserving gates. The invariants impose thefollowing restrictions on a typical block ( a b

c d ).

X -preserving: b = 0,

Y -preserving: a + b + c + d = 0,

Z -preserving: c = 0.

But CNOT is X - and Z -preserving! 1 0 0 00 1 0 1

1 0 1 00 0 0 1

Tour of the Lattice

C(X , X ) + X C(Y , Y ) + Y C(Z , Z) + Z

T4 + P + Γ

T4 + P

Tour of the Lattice

C(X , X ) + X C(Y , Y ) + Y C(Z , Z) + Z

T4 + P + Γ

T4 + P

Tour of the Lattice

C(X , X ) + X C(Y , Y ) + Y C(Z , Z) + Z

T4 + P + Γ

T4 + P

Tour of the Lattice

T4 gate

Definition

Let T4 be the 4 qubit gate which

does nothing to even-parity inputs,

flips all bits of odd-parity inputs.

E.g. T4|0000〉 = |0000〉, and T4|0001〉 = |1110〉.

Tour of the Lattice

T4 tableau

The tableau for T4:

0 0 1 0 1 0 1 00 0 0 1 0 1 0 1

1 0 0 0 1 0 1 00 1 0 0 0 1 0 1

1 0 1 0 0 0 1 00 1 0 1 0 0 0 1

1 0 1 0 1 0 0 00 1 0 1 0 1 0 0

Amazingly, T4 is X -, Y -, and Z -preserving!

Tour of the Lattice

Symmetry?

C(X , X ) + X C(Y , Y ) + Y C(Z , Z) + Z

T4 + P + Γ

T4 + P

Tour of the Lattice

Three-fold symmetry

The automorphism of the Pauli group that maps

I → I

X → Y

Y → Z

Z → X ,

lifts to an automorphism of the Clifford group, and an automorphism of the lattice.

Tour of the Lattice

Z-preserving and misc. classes

T4 + P + Γ CNOT +P + S

CNOT +P T4 + P + S

T4 + P

CSIGN +P + S

CSIGN +P CSIGN +S CSIGN +Z + θXY

CSIGN +Z

Proof Techniques

Section 4

Proof Techniques

Steps of the proof:

1 Characterize each class by generators and an invariant.

E.g., T4 generates all gates that are simultaneously X -, Y -, and Z -preserving.

2 Argue that the classes are distinct.

E.g., CNOT is not in the class generated by T4 because CNOT is not Y -preserving.

3 Argue that no class has been omitted.

Proof Techniques

Steps of the proof:

Proof Techniques

Steps of the proof:

Proof Techniques

Steps of the proof:

Proof Techniques

Steps of the proof:

Proof Techniques

Steps of the proof:

Proof Techniques

No missing classes

1 Suppose for a contradiction that 〈G 〉 is not a listed class.

2 For each invariant G does not satisfy, construct a simple gate violating that invariant.E.g., if some gate is not Y -preserving, extract one of

a single qubit gate that is not Y -preserving,a CSIGN,a CNOT, etc.

3 Use simple gates to construct canonical class generators.

E.g., the class is defined as generated by T4 and S , but you have T4 and CSIGN.

Proof Techniques

No missing classes

1 Suppose for a contradiction that 〈G 〉 is not a listed class.2 For each invariant G does not satisfy, construct a simple gate violating that invariant.

E.g., if some gate is not Y -preserving, extract one of

Proof Techniques

No missing classes

1 Suppose for a contradiction that 〈G 〉 is not a listed class.2 For each invariant G does not satisfy, construct a simple gate violating that invariant.

E.g., if some gate is not Y -preserving, extract one of

Proof Techniques

1 Identify 2× 2 block of the tableau violating the invariant....

· · · 0 11 0 · · ·...

2 Apply a SWAP to make that row the first one. · · · 0 1

1 0 · · ·...

Proof Techniques

Assume the first row is (a1 a2 . . . a2k b1 . . . b` 0 . . . 0

)where

a1, . . . , a2k are invertible 2× 2 blocks,

b1, . . . , b` 6= 0 are non-invertible 2× 2 blocks.

Proof Techniques

Universal Construction

......

U U−1

Proof Techniques

b1 b2. . . b` b` . . . b2 b1

G(a2) G−1(a2)

G(a3) G−1(a3)

......

G(a2k) G−1(a2k)

b∗1 b∗1

b∗2 b∗2

. . . . ..

b∗` b∗`

Open Problems

Section 5

Open Problems

Complete the classification of quantum gates. Possible strategy: divide into cases bygroup of single qubit gates.

Classify Clifford gates but with stabilizer state ancillas. We can show nothing changesunless

〈Γ⊗ Γ−1〉 6= 〈Γ〉,

where Γ = HS .

Open Problems

Complete the classification of quantum gates. Possible strategy: divide into cases bygroup of single qubit gates.

Classify Clifford gates but with stabilizer state ancillas. We can show nothing changesunless

〈Γ⊗ Γ−1〉 6= 〈Γ〉,

where Γ = HS .

the classification of clifford gates over qubits

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